In honor of DUJS’s rich tradition of poetry, started by former Public Relations Officer Ed Chien ’09.
The concept of an infinitesimal is more easily illustrated than directly defined,
So first, here is a simple demonstration that you can run quickly through your mind:
You and I have a delectable cake, and we decide to split it in two.
You randomly suggest we continue halving the sugary confection, and so we merrily do.
An infinite number of cuts later, we are essentially left with infinitesimal pieces of cake,
That is, we are left with miniscule morsels that we cannot define as cake, but saying that they are nothing would be a huge mistake.
Now replace the cake with the set of all real numbers, large and small,
And perhaps this childish prelude will appear somewhat mathematical after all.
Infinitesimals are numbers greater than zero with absolute values less than all positive reals, to be more mathematically precise.
Infinitesimals are infinitely small numbers, to be more mathematically concise (1).
For instance, the y values of graph y = 1/x as x → infinity,
Or the value of ∆x in the limit definition of derivative as ∆x enters zero’s vicinity.
Applied now to some areas of math, economics, and physics, infinitesimals have gained some fame,
But they are perhaps best known for the relatively new variety of calculus that now bears their name.
Infinitesimals would appear indispensible to such basic calculus concepts as the derivative, yes,
But their acceptance has been a struggle nevertheless.
The concept of the infinitesimal first appeared in the method of exhaustion, a roundabout precursor to limits from Ancient Greece.
Archimedes used this method to find areas and volumes, though short of an acknowledgement of infinitesimals his progress did cease (2).
Next Newton, co-developer of calculus, attempted to bring infinitesimals into the spotlight.
His “fluxions” the former name for derivative or instantaneous change – never really took flight (3).
Leibnitz, other father of calculus, thought, too, that the concept of infinitesimals must exist.
These “ideal numbers” would be infinitely small but with properties of reals, he did insist (4).
But you have no rigorous definition or proof, to Newton and Leibnitz other mathematicians did say.
And ‘twas true that neither was able to describe these numbers-approaching-zero in anything more than an intuitive way.
Thus, for a couple hundred years, at the idea of infinitesimals mathematicians shook a collective head.
And when Weierstrass presented his epsilon-delta definition of limits, they all adopted this instead (4).
In the 1900s, the quest to formalize the infinitesimal did once more commence.
Hahn in 1907 and Laugwitz and Schmieden in the 1950s came to the infinitesimal’s defense (4).
Then finally in the 1960s, there was a major break-through.
Abraham Robinson showed that infinitesimals are real, too (2)!
Well technically, Robinson said that infinitesimals are hyperreal, not real.
And, as Leibnitz prophesized, to the properties of real numbers they do indeed appeal.
Formally, hyperreal numbers encompass all reals, as well as all numbers infinitely close to reals – as close as you can get.
Hyperreal numbers infinitely close to zero – infinitesimals is what we call this subset (2).
And so, rigorous definition achieved, Team Infinitesimal their battle had won.
A new era of “non-standard analysis,” as Robinson dubbed it, seemed to have begun (5).
Since, infinitesimals have found their way into research and some calculus textbooks
Though some question whether the concept, still lacking widespread use, is as revolutionary as it looks (6).
Acknowledgements
Special thanks are due to Dr. Lu-Chang Qin, University of North Carolina at Chapel Hill, for bringing to light a most excellent analogy between cake and infinitesimals.
References
1. H. J. Keisler, Foundations of Infinitesimal Calculus (2009). Available at http://www.math.wisc.edu/~keisler/foundations.pdf (10 Aug 2009).
2. M. Hazewinkel, Encyclopedia of Mathematics (Springer-Verlag, Berlin 2002).
3. Encylopaedia Britannica, fluxion (2009). Available at http://www.britannica.com/EBchecked/topic/211566/fluxion (10 Aug 2009).
4. K. D. Stroyan, Introduction to the Theory of Infinitesimals (Academic Press, Missouri 1976).
5. A. Robinson, Non-standard Analysis (Princeton University Press, New Jersey 1996).
6. E. Bishop, Bulletin of the American Mathematical Society, 38(2) (American Mathematical Society, Rhode Island 1977).